Al-Salam–Carlitz polynomials

In mathematics, Al-Salam–Carlitz polynomials U$(a) n$(x;q) and V$(a) n$(x;q) are two families of basic hypergeometric orthogonal polynomials in the basic Askey scheme, introduced by. give a detailed list of their properties.

Definition
The Al-Salam–Carlitz polynomials are given in terms of basic hypergeometric functions by
 * $$ U_n^{(a)}(x;q) = (-a)^nq^{n(n-1)/2}{}_2\phi_1(q^{-n}, x^{-1};0;q,qx/a)$$
 * $$ V_n^{(a)}(x;q) = (-a)^nq^{-n(n-1)/2}{}_2\phi_0(q^{-n}, x;-;q,q^n/a)$$